Precalculus Examples

${(3 u^{3} v^{3})}^{2} {(2 u^{- 2} v)}^{3}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1

Apply the product rule to $3 u^{3} v^{3}$ .

${(3 u^{3})}^{2} {(v^{3})}^{2} {(2 u^{- 2} v)}^{3}$

Step 1.2

Apply the product rule to $3 u^{3}$ .

$3^{2} {(u^{3})}^{2} {(v^{3})}^{2} {(2 u^{- 2} v)}^{3}$

Step 2

Raise $3$ to the power of $2$ .

$9 {(u^{3})}^{2} {(v^{3})}^{2} {(2 u^{- 2} v)}^{3}$

Step 3

Multiply the exponents in ${(u^{3})}^{2}$ .

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Step 3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$9 u^{3 \cdot 2} {(v^{3})}^{2} {(2 u^{- 2} v)}^{3}$

Step 3.2

Multiply $3$ by $2$ .

$9 u^{6} {(v^{3})}^{2} {(2 u^{- 2} v)}^{3}$

Step 4

Multiply the exponents in ${(v^{3})}^{2}$ .

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Step 4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$9 u^{6} v^{3 \cdot 2} {(2 u^{- 2} v)}^{3}$

Step 4.2

Multiply $3$ by $2$ .

$9 u^{6} v^{6} {(2 u^{- 2} v)}^{3}$

Step 5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$9 u^{6} v^{6} {(2 \frac{1}{u^{2}} v)}^{3}$

Step 6

Combine $2$ and $\frac{1}{u^{2}}$ .

$9 u^{6} v^{6} {(\frac{2}{u^{2}} v)}^{3}$

Step 7

Combine $\frac{2}{u^{2}}$ and $v$ .

$9 u^{6} v^{6} {(\frac{2 v}{u^{2}})}^{3}$

Step 8

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.1

Apply the product rule to $\frac{2 v}{u^{2}}$ .

$9 u^{6} v^{6} \frac{{(2 v)}^{3}}{{(u^{2})}^{3}}$

Step 8.2

Apply the product rule to $2 v$ .

$9 u^{6} v^{6} \frac{2^{3} v^{3}}{{(u^{2})}^{3}}$

Step 9

Simplify the expression.

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Step 9.1

Raise $2$ to the power of $3$ .

$9 u^{6} v^{6} \frac{8 v^{3}}{{(u^{2})}^{3}}$

Step 9.2

Multiply the exponents in ${(u^{2})}^{3}$ .

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Step 9.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$9 u^{6} v^{6} \frac{8 v^{3}}{u^{2 \cdot 3}}$

Step 9.2.2

Multiply $2$ by $3$ .

$9 u^{6} v^{6} \frac{8 v^{3}}{u^{6}}$

Step 10

Cancel the common factor of $u^{6}$ .

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Step 10.1

Factor $u^{6}$ out of $9 u^{6} v^{6}$ .

$u^{6} (9 v^{6}) \frac{8 v^{3}}{u^{6}}$

Step 10.2

Cancel the common factor.

$u^{6} (9 v^{6}) \frac{8 v^{3}}{u^{6}}$

Step 10.3

Rewrite the expression.

$9 v^{6} (8 v^{3})$

Step 11

Multiply $8$ by $9$ .

$72 v^{6} v^{3}$

Step 12

Multiply $v^{6}$ by $v^{3}$ by adding the exponents.

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Step 12.1

Move $v^{3}$ .

$72 (v^{3} v^{6})$

Step 12.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$72 v^{3 + 6}$

Step 12.3

Add $3$ and $6$ .

$72 v^{9}$